# Show this is a projection on a vector space

• JaysFan31

## Homework Statement

Let V=Mn(F) be the space of all nxn matrices over F; define TA=(1/2)(A+transpose(A)) for A in V.
Verify that T is not only a linear operator on V, but is also a projection.

## Homework Equations

A is a projection when A squared=A.

## The Attempt at a Solution

I don't see how this works since clearly (1/2)(A+transpose(A)) squared does not equal (1/2)(A+transpose(A)) for all matrices.

What am I doing wrong?

A is a projection when A2 = A, not when (Ax)2 = Ax. So you don't need to look at whether

$$\left [\frac{1}{2}(A + A^t)\right ]^2 = \frac{1}{2}(A + A^t)$$

You need to look at whether T2 = T, i.e. whether T(TA) = TA for all A, i.e. whether:

$$\frac{1}{2}\left [\left (\frac{1}{2}(A + A^t)\right ) + \left (\frac{1}{2}(A + A^t)\right )^t\right ] = \frac{1}{2}(A + A^t)$$

Remember, you're used to writing A for your linear operators, and vectors in your vector space V are normally written as x or v or something. But now you have matrices AS THE VECTORS IN YOUR VECTOR SPACE, so you'll probably use A to stand for a vector, and now T is the operator.

And you still need to check linearity.